How can a evenly spaced grid with a circular boundary be modeled?
Grid first, then cut it by Knife Projecting a circle.

I would start by modeling the evenly spaced grid with a couple of Array modifier (one to repeat the geometry on the X axis and one on the Y). Check the Merge boxes to avoid obtain disconnected geometry.



Apply the two modifiers, then create the circle and Knife project it on the grid.



Invert selection and delete the unwanted faces. Now Inset the faces individually, Extrude, Mirror the result, and so on...



In my example I also Beveled the edges and added a Lattice modifier after some Simple Subdivison iterations as you can see in the modifier tab on the right.





Bonus:

You can loopcut a circle's faces if you model them properly (no tris, axial edge flow) as shown in the gif below:



but it would be trickyer to get an evely spaced grid because of the vertices distribution along the circle. That's why is better to model the grid first.